# Properties

 Label 8-825e4-1.1-c1e4-0-6 Degree $8$ Conductor $463250390625$ Sign $1$ Analytic cond. $1883.32$ Root an. cond. $2.56664$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3-s + 2·4-s − 2·6-s + 8·7-s − 5·8-s + 4·11-s + 2·12-s + 4·13-s − 16·14-s + 5·16-s − 2·17-s − 5·19-s + 8·21-s − 8·22-s + 14·23-s − 5·24-s − 8·26-s + 16·28-s − 5·29-s − 7·31-s + 2·32-s + 4·33-s + 4·34-s − 7·37-s + 10·38-s + 4·39-s + ⋯
 L(s)  = 1 − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 3.02·7-s − 1.76·8-s + 1.20·11-s + 0.577·12-s + 1.10·13-s − 4.27·14-s + 5/4·16-s − 0.485·17-s − 1.14·19-s + 1.74·21-s − 1.70·22-s + 2.91·23-s − 1.02·24-s − 1.56·26-s + 3.02·28-s − 0.928·29-s − 1.25·31-s + 0.353·32-s + 0.696·33-s + 0.685·34-s − 1.15·37-s + 1.62·38-s + 0.640·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{8} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$1883.32$$ Root analytic conductor: $$2.56664$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{825} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.673108267$$ $$L(\frac12)$$ $$\approx$$ $$2.673108267$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
5 $$1$$
11$C_4$ $$1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
good2$C_2^2:C_4$ $$1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
7$C_4\times C_2$ $$1 - 8 T + 17 T^{2} + 50 T^{3} - 299 T^{4} + 50 p T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2:C_4$ $$1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
17$C_4\times C_2$ $$1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 60 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
19$C_4\times C_2$ $$1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 65 p T^{5} + 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2^2:C_4$ $$1 + 5 T + 11 T^{2} + 195 T^{3} + 1736 T^{4} + 195 p T^{5} + 11 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
31$C_4\times C_2$ $$1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 91 p T^{5} + 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2:C_4$ $$1 + 7 T + 32 T^{2} + 365 T^{3} + 3451 T^{4} + 365 p T^{5} + 32 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2:C_4$ $$1 + 17 T + 68 T^{2} - 661 T^{3} - 8425 T^{4} - 661 p T^{5} + 68 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 - 13 T + p T^{2} - 325 T^{3} + 4016 T^{4} - 325 p T^{5} + p^{3} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2:C_4$ $$1 - 9 T - 7 T^{2} + 525 T^{3} - 3884 T^{4} + 525 p T^{5} - 7 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
59$C_4\times C_2$ $$1 + 15 T + 76 T^{2} + 675 T^{3} + 8161 T^{4} + 675 p T^{5} + 76 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
61$C_4\times C_2$ $$1 + 7 T - 12 T^{2} - 511 T^{3} - 2845 T^{4} - 511 p T^{5} - 12 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 + 21 T + 243 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_4\times C_2$ $$1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 624 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2:C_4$ $$1 + T + 68 T^{2} + 155 T^{3} + 5911 T^{4} + 155 p T^{5} + 68 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^2:C_4$ $$1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2^2:C_4$ $$1 - 14 T + 53 T^{2} - 870 T^{3} + 14801 T^{4} - 870 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2:C_4$ $$1 + 17 T + 12 T^{2} - 1445 T^{3} - 15649 T^{4} - 1445 p T^{5} + 12 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$